A, B and C can complete a job working alone in 6 days, 9 days and 18 days respectively. They all work together for 1 day and then only C continues working alone. How many additional days will C take to finish the remaining work?

Introduction / Context:
This question is another variation on combined work followed by a single worker completing the remaining part. Knowing the individual times for A, B, and C, we first find how much work is completed when all three work together for one day, and then use C's individual rate to determine the extra time needed. Problems of this type reinforce the idea of breaking a job into fractions completed per day.

Given Data / Assumptions:

• A alone can complete the job in 6 days.
• B alone can complete the job in 9 days.
• C alone can complete the job in 18 days.
• All three work together for 1 day from the start.
• After that, A and B stop, and C finishes the rest alone.
• All work rates are constant.

Concept / Approach:
As usual, we consider the total work to be 1 unit. We convert each worker's time into a daily work rate. Adding the three rates gives the joint work done in a day when they are all working. Subtracting that from 1 yields the remaining fraction of the work. Dividing this remainder by C's individual rate gives the extra days C needs to complete the job alone.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: A's rate = 1/6 of the work per day. Step 3: B's rate = 1/9 of the work per day. Step 4: C's rate = 1/18 of the work per day. Step 5: Combined rate of A + B + C = 1/6 + 1/9 + 1/18. Step 6: Using common denominator 18, this becomes 3/18 + 2/18 + 1/18 = 6/18 = 1/3. Step 7: Work done in the first day by all three = 1/3 of the job. Step 8: Remaining work = 1 − 1/3 = 2/3 of the job. Step 9: C alone works at 1/18 per day, so time required = (2/3) / (1/18) = (2/3) × 18 = 12 days.

Verification / Alternative check:
We can check approximate logic. C alone would need 18 days to complete the full job. Here, two-thirds of the job remains when C starts alone. Two-thirds of 18 is 12, which matches the exact calculation. Also, the total work done is 1/3 in the first day plus 2/3 over the next 12 days, summing to 1. Everything is consistent with the given data.

Why Other Options Are Wrong:
9 and 10 days are too small; at 1/18 per day, in 9 days C would complete only half the job (9/18), not two-thirds. 6 days is even smaller. 15 days is larger than the two-thirds scaled time of 18, and would result in more than the remaining work being done. Thus only 12 days fits both the rate and the remaining fraction of the job.

Common Pitfalls:
One pitfall is forgetting that the first day of combined work is only 1 day, not the entire duration. Another is incorrectly adding the fractions for the combined rate. Some learners also try to directly use proportions of days instead of rates, which leads to incorrect conclusions. Keeping track of rates and the fraction of work completed at each step prevents these errors.

Final Answer:
C will take 12 additional days to finish the remaining work.